In his recent work [1]} Polishchuk showed that non-degenerate skew-symmetric solutions of AYBE satisfying (REF ) can be obtained from appropriate triple Massey products in the perfect derived category of coherent sheaves \(\operatorname{\mathsf {Perf}}(\mathbb {E})\)  on a non-commutative projective curve \(\mathbb {E}= (E, \mathcal {A})\) , where \(E\)  is an irreducible projective curve over \(\mathbb {C}\)  of arithmetic genus one and \(\mathcal {A}\)  is a symmetric spherical order on \(E\) .
A simplest example of such an order is given by \(\mathcal {A}= \mbox{\emph {End}}_E(\mathcal {F})\) , where \(\mathcal {F}\)  is a simple vector bundle on \(E\) . Let \(E = E_\tau := \mathbb {C}/\langle 1, \tau \rangle \)  be the elliptic curve determined by \(\tau \in \mathbb {C}\)  and \(\mathcal {F}\)  be a simple vector bundle of rank \(n\)  and degree \(d\)  on \(E\) . It follows from results of Atiyah [2]} that such \(\mathcal {F}\)  exists and the sheaf of algebras \(\mathcal {A}= \mathcal {A}_{(n, d)} := \mbox{\emph {End}}_E(\mathcal {F})\)  does not depend on the choice of \(\mathcal {F}\) . We show that the solution of AYBE arising from the non-commutative projective curve \(\bigl (E_\tau , \mathcal {A}_{(n, d)}\bigr )\)  is given by the formula (REF ). In [3]}, the corresponding computations were performed using the homological mirror symmetry and explicit formulae for triple Massey products in the Fukaya category of a torus. The expression for the resulted solution of AYBE (see [3]}) was different from (REF ). Our computations are straightforward and based by techniques developed in the articles [5]}, [6]}.
